What is the Altitude of a Triangle?
The altitude (or height) of a triangle is a perpendicular line segment drawn from a vertex to the line containing the opposite side (the base). Every triangle has three altitudes — one from each vertex. This calculator finds the height of a triangle using five different methods, supporting both metric (mm, cm, m) and US imperial (in, ft, yd) units.
How to Find the Height of a Triangle — Formulas
1. Given Area and Base
The most straightforward formula comes from the basic area equation:
Area = ½ × base × height
Solving for height: h = 2 × Area / base
2. Given Three Sides (Heron's Formula)
When only the three side lengths a, b, c are known, use Heron's formula to find the area first:
s = (a + b + c) / 2
Area = √(s × (s−a) × (s−b) × (s−c))
Then compute each altitude:
h_a = 2 × Area / a, h_b = 2 × Area / b, h_c = 2 × Area / c
This method works for any scalene triangle and gives all three altitudes at once.
3. How to Find the Height of an Equilateral Triangle
An equilateral triangle has all three sides equal (a). Its altitude is:
h = (√3 / 2) × a ≈ 0.866 × a
Because all sides are equal, all three altitudes are identical.
4. How to Find the Height of an Isosceles Triangle
For an isosceles triangle with equal legs a and base b, the altitude from the apex bisects the base:
h = √(a² − (b/2)²)
This is derived directly from the Pythagorean theorem applied to the right triangle formed by the altitude, half the base, and one leg.
5. How to Find the Altitude of a Right Triangle
For a right triangle with legs a and b, the altitude drawn from the right angle to the hypotenuse c is:
c = √(a² + b²), then h = (a × b) / c
This altitude is also the geometric mean of the two segments it creates on the hypotenuse.
Frequently Asked Questions
Can a triangle have more than one height?
Yes. Every triangle has exactly three heights (altitudes), one corresponding to each side as the base. This calculator computes the altitude to the selected base.
What units does this calculator support?
The calculator supports metric units (millimeters, centimeters, meters) and US imperial units (inches, feet, yards). The result is expressed in the same unit as the input.
What is "dropping the altitude"?
When we say we "drop" an altitude, it means we draw a perpendicular from a vertex to the opposite side (or its extension). The foot of this perpendicular is where the altitude meets the base.
Is the altitude always inside the triangle?
For an acute triangle, all three altitudes lie inside the triangle. For a right triangle, two altitudes coincide with the legs. For an obtuse triangle, two altitudes fall outside the triangle (the perpendiculars meet extensions of the sides).
How is the height related to the area?
The relationship is always: Area = ½ × base × height. Knowing any two of the three values (area, base, height) lets you find the third.